Let $F=P / Q$ be a rational function, where $\operatorname{deg} Q \geqslant \operatorname{deg} P+2$ and $Q$ has no real zeros. Using the calculus of residues, write a general expression for

$$\int_{-\infty}^{\infty} F(x) e^{i x} d x$$

in terms of residues and briefly sketch its proof.

Evaluate explicitly the integral

$$\int_{-\infty}^{\infty} \frac{\cos x}{4+x^{4}} d x$$